The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A084231 Numbers k such that the root-mean-square value of 1, 2, ..., k, i.e., sqrt((1/k)*Sum_{j=1..k} j^2), is an integer. 6
1, 337, 65521, 12710881, 2465845537, 478361323441, 92799630902161, 18002650033695937, 3492421306906109761, 677511730889751597841, 131433783371304903871537, 25497476462302261599480481, 4946378999903267445395341921, 959572028504771582145096852337 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Equivalently, numbers k such that sqrt((k+1)*(2*k+1)/6) is an integer.
LINKS
Tanya Khovanova, Recursive Sequences
FORMULA
a(n) = ((7/2 + 2*sqrt(3))*(97 + 56*sqrt(3))^n + (7/2 - 2*sqrt(3))*(97 - 56*sqrt(3))^n - 3)/4.
a(n) = (floor((7/2 + 2*sqrt(3))*(97 + 56*sqrt(3))^n) - 2)/4.
a(n+3) = 195*(a(n+2) - a(n+1)) + a(n).
G.f.: x*(1+142*x+x^2)/((1-x)*(1-194*x+x^2)).
a(n) = ((7 - 4*sqrt(3))^(1+2n) + (7 + 4*sqrt(3))^(1+2n) - 6)/8. - Peter Pein (peter.pein(AT)dordos.de), Mar 03 2005
a(n) = 195*a(n-1) - 195*a(n-2) + a(n-3), with a(0)=0, a(1)=1, a(2)=337, a(3)=65521. - Harvey P. Dale, Jul 14 2011
EXAMPLE
337 is in the sequence because sqrt((1/337)*Sum_{k=1..337} k^2) is an integer (195=A084232(1)).
MATHEMATICA
a[n_]:=Expand[((7-4 Sqrt[3])^(1+2n)+(7+4 Sqrt[3])^(1+2n)-6)/8] (* Peter Pein *)
CoefficientList[Series[x (1+142x+x^2)/((1-x)(1-194x+x^2)), {x, 0, 30}], x] (* or *) Join[{0}, LinearRecurrence[{195, -195, 1}, {1, 337, 65521}, 30]] (* Harvey P. Dale, Jul 14 2011 *)
CROSSREFS
Cf. A084232.
Sequence in context: A263865 A184202 A194478 * A243483 A234625 A226539
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
One more term from Peter Pein (peter.pein(AT)dordos.de), Mar 03 2005
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 14 21:35 EDT 2024. Contains 373401 sequences. (Running on oeis4.)